Orthogonal sums of semigroups (Q1895088)

A subset \(A\) of a semigroup \(S\) with zero is called 0-consistent if for every \(x, y \in S\), \(xy \in A\setminus \{0\}\) implies \(x, y \in A\). Every semigroup with zero is an orthogonal sum of orthogonal indecomposable semigroups and the set of all 0-consistent ideals is a complete Boolean algebra whose atoms are summands in the greatest orthogonal decomposition of the semigroup.